A047924 - OEIS

%I #37 Aug 25 2022 17:31:45

%S 3,6,11,14,19,24,27,32,35,40,45,48,53,58,61,66,69,74,79,82,87,90,95,

%T 100,103,108,113,116,121,124,129,134,137,142,147,150,155,158,163,168,

%U 171,176,179,184,189,192,197,202,205,210,213,218,223,226,231,234,239

%N a(n) = B_{A_n+1}+1, where A_n = floor(n*phi) = A000201(n), B_n = floor(n*phi^2) = A001950(n) and phi is the golden ratio.

%C 2nd column of array in A038150.

%C Apart from the first term also the second column of A126714; see also A223025. - _Casey Mongoven_, Mar 11 2013

%D Clark Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995), 129-138.

%H Vincenzo Librandi, <a href="/A047924/b047924.txt">Table of n, a(n) for n = 0..1000</a>

%H A. S. Fraenkel, <a href="https://doi.org/10.1016/S0304-3975(00)00062-1">Recent results and questions in combinatorial game complexities</a>, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.

%H A. S. Fraenkel, <a href="https://doi.org/10.1016/S0304-3975(01)00070-6">Arrays, numeration systems and Frankenstein games</a>, Theoret. Comput. Sci. 282 (2002), 271-284; <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/ans1.ps">preprint</a>.

%H Clark Kimberling, <a href="http://www.fq.math.ca/Scanned/32-4/kimberling.pdf">The first column of an interspersion</a>, The Fibonacci Quarterly 32 (1994), 301-315.

%p A001950 := proc(n)

%p         local phi;

%p         phi := (1+sqrt(5))/2 ;

%p         floor(n*phi^2) ;

%p end proc:

%p A000201 := proc(n)

%p         local phi;

%p         phi := (1+sqrt(5))/2 ;

%p         floor(n*phi) ;

%p end proc:

%p A047924 := proc(n)

%p         1+A001950(1+A000201(n)) ;

%p end proc: # _R. J. Mathar_, Mar 20 2013

%t A[n_] := Floor[n*GoldenRatio]; B[n_] := Floor[n*GoldenRatio^2]; a[n_] := B[A[n]+1]+1; Table[a[n], {n, 0, 56}] (* _Jean-François Alcover_, Feb 11 2014 *)

%o (Python)

%o from mpmath import *

%o mp.dps=100

%o import math

%o def A(n): return int(math.floor(n*phi))

%o def B(n): return int(math.floor(n*phi**2))

%o def a(n): return B(A(n) + 1) + 1 # _Indranil Ghosh_, Apr 25 2017

%o (Python)

%o from math import isqrt

%o def A047924(n): return ((m:=(n+isqrt(5*n**2)>>1)+1)+isqrt(5*m**2)>>1)+m+1 # _Chai Wah Wu_, Aug 25 2022

%Y Cf. A007066.

%K nonn,nice,easy

%O 0,1

%A _N. J. A. Sloane_

%E More terms from _Naohiro Nomoto_, Jun 08 2001

%E New description from _Aviezri S. Fraenkel_, Aug 03 2007